In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations (e.g., + - × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit. The set of operations and functions admitted in a closed-form expression may vary with author and context.
The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation
is tractable since its solutions can be expressed as a closed-form expression, i.e. in terms of elementary functions:
Similarly solutions of cubic and quartic (third and fourth degree) equations can be expressed using arithmetic, square roots, and cube roots, or alternatively using arithmetic and trigonometric functions. However, there are quintic equations without closed-form solutions using elementary functions, such as x5 - x + 1 = 0.
An area of study in mathematics referred to broadly as Galois theory involves proving that no closed-form expression exists in certain contexts, based on the central example of closed-form solutions to polynomials.
Changing the definition of "well-known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well-known since numerical implementations are widely available.
An analytic expression (or expression in analytic form) is a mathematical expression constructed using well-known operations that lend themselves readily to calculation. Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the basic arithmetic operations (addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the nth root), logarithms, and trigonometric functions.
However, the class of expressions considered to be analytic expressions tends to be wider than that for closed-form expressions. In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.
If an analytic expression involves only the algebraic operations (addition, subtraction, multiplication, division, and exponentiation to a rational exponent) and rational constants then it is more specifically referred to as an algebraic expression.
Closed-form expressions are an important sub-class of analytic expressions, which contain a bounded or an unbounded number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include infinite series or continued fractions; neither includes integrals or limits. Indeed, by the Stone-Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.
Similarly, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed as a closed-form expression; and it is said to have an analytic solution if and only if at least one solution can be expressed as an analytic expression. There is a subtle distinction between a "closed-form function" and a "closed-form number" in the discussion of a "closed-form solution", discussed in (Chow 1999) and below. A closed-form or analytic solution is sometimes referred to as an explicit solution.
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|Arithmetic expressions||Polynomial expressions||Algebraic expressions||Closed-form expressions||Analytic expressions||Mathematical expressions|
|Elementary arithmetic operation||Yes||Addition, subtraction, and multiplication only||Yes||Yes||Yes||Yes|
|Finite continued fraction||Yes||No||Yes||Yes||Yes||Yes|
|Integer nth root||No||No||Yes||Yes||Yes||Yes|
|Inverse trigonometric function||No||No||No||Yes||Yes||Yes|
|Inverse hyperbolic function||No||No||No||Yes||Yes||Yes|
|Non-algebraic root of a polynomial||No||No||No||No||Yes||Yes|
|Gamma function and factorial of a non-integer||No||No||No||No||Yes||Yes|
|Infinite sum (series) (including power series)||No||No||No||No||Convergent only||Yes|
|Infinite product||No||No||No||No||Convergent only||Yes|
|Infinite continued fraction||No||No||No||No||Convergent only||Yes|
The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to as differential Galois theory, by analogy with algebraic Galois theory.
A standard example of an elementary function whose antiderivative does not have a closed-form expression is:
whose antiderivative is (up to constants) the error function:
Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouville numbers, EL numbers and elementary numbers. The Liouville numbers, denoted L (not to be confused with Liouville numbers in the sense of rational approximation), form the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)--that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in (Ritt 1948, p. 60). L was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in (Chow 1999, pp. 441-442), denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm--this need not be algebraically closed, and correspond to explicit algebraic, exponential, and logarithmic operations. "EL" stands both for "Exponential-Logarithmic" and as an abbreviation for "elementary".
Whether a number is a closed-form number is related to whether a number is transcendental. Formally, Liouville numbers and elementary numbers contain the algebraic numbers, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via transcendental number theory, in which a major result is the Gelfond-Schneider theorem, and a major open question is Schanuel's conjecture.
For purposes of numeric computations, being in closed form is not in general necessary, as many limits and integrals can be efficiently computed.
There is software that attempts to find closed-form expressions for numerical values, including RIES,identify in Maple and SymPy, Plouffe's Inverter, and the Inverse Symbolic Calculator.